How Many Grapes Did Sophia Eat? A Detailed Math Problem Explained

Let’s dive into a fun and engaging mathematical problem featuring Sophia and her grape bag. This problem involves understanding fractions, proportions, and logical reasoning. Follow along to discover how we can solve this intriguing puzzle step-by-step.

Introduction to the Problem

Sophia has a bag containing red and green grapes. Initially, 4/9 of the grapes are red. After Sophia picks and eats one red grape, the remaining grapes have a 3/7 proportion of red grapes. We need to figure out the total number of grapes, how many are red, and how many are green initially and finally.

Setting Up the Problem

Let the total number of grapes in Sophia's bag be x.

The number of red grapes initially is: (frac{4}{9}x) The number of green grapes initially is: (x - frac{4}{9}x frac{5}{9}x)

After eating one red grape:

The number of red grapes left is: (frac{4}{9}x - 1) The total number of grapes left is: (x - 1) According to the problem, 3/7 of the remaining grapes are red:

(frac{3}{7}(x-1) frac{4}{9}x - 1)

Solving the Equation

First, let's simplify and solve for x:

Eliminate the fractions by multiplying both sides by the least common multiple (LCM) of 7 and 9, which is 63: (63 cdot frac{3}{7}(x-1) 63 cdot (frac{4}{9}x - 1)) This simplifies to: (27(x-1) 28x - 63) Distribute and simplify: (27x - 27 28x - 63) Rearrange to isolate x: (-x -36) Solving for x we get: x 36

Calculating Initial Counts

Now, let's determine the number of red and green grapes initially:

The number of red grapes is: (frac{4}{9} cdot 36 16) The number of green grapes is: (frac{5}{9} cdot 36 20)

After Sophia eats one red grape:

The number of red grapes left is: (16 - 1 15) The total number of grapes left is: (36 - 1 35) Checking the proportion of red grapes remaining: (frac{3}{7} cdot 35 15)

Thus, the number of remaining grapes in the bag is:

Red grapes: 15 Green grapes: 20

Conclusion

Through this detailed step-by-step process, we have solved the mathematical problem and determined that initially, Sophia had 36 grapes, 16 of which were red, and 20 were green. After eating one red grape, there are 15 red grapes and 35 total grapes left, with the remaining 20 grapes being green.

This problem not only tests mathematical skills but also reinforces the importance of logical reasoning and problem-solving techniques in everyday situations.